As we learn in the Orthodox Church, the vast majority of real Saints do not get involved in politics. Many of them withdraw from the world in monasteries or caves.
However, a few Saints are Warriors and national leaders like S. Alexander Nevsky and Sts. Vladimir and Constantine.. Others choose monasticism, but are called by their people into a leadership role, like St. Sava of Serbia.
Most of us are laypeople who are called to engage with the world, including the dirty, messy business of politics and of course of employment. But some monastic Saints taught that many laypeople are more saintly than most monastics, despite our engagement in the messy business of the world. (So there is hope for even me, a sinner!!!)
However, whatever our calling, we are reminded that our true citizenship is in the Kingdom of God.
Such a tendency would militate against those starkly intended in a pure life from even being seen among the political world.
Maybe this is a mistake. The Protestant evangelical model would aver that it is a mistake, that it’s better to stay out in public (if possible) and serve as a conduit of blessing from God to other people. I.e. the career monk or nun is likely being more idealistic than even God is.
Though understood here, the Orthodox sector of the church has an answer as you mentioned, that there is a public demand for a holy man (or woman) to come out of the monastic life and help a public situation.
In a modern democratic republic, this may or may not be hard to envision. Perhaps a wise man like Donald Trump could end up getting a reference to someone in a monastery, but I’m thinking more likely he’d just respect this person’s vocation and go elsewhere.
I want to live a Christian life, but the more I approach it, the more I see that it looks like the public is the place that makes the most sense to live it in. But that might just be my calling. Some people might be so put upon by the demands of a public life that a monastic life would make more sense. God has a plan for every willingly yielding person.